NOTATION, SIMPLE NUMBERS, TABLE OF DENOMINATE NUMBERS, ETC. DEFINITIONS. Mathematics is the science of quantity. Quantity is anything that can be increased, diminished, or measured. Arithmetic is founded on notation, and its operations are carried out by means of addition, subtraction, multiplication, and division. Notation is a method of writing or expressing numbers by characters. Numeration is a method of reading numbers expressed by characters. A Number is a unit or a collection of units. A Unit is a single thing such as 1, a boiler, a ship, etc. ARABIC SYSTEM OF NOTATION. Numbers are written by words, letters, and figures. In the Arabic system the numbers are expressed by the position of the ten characters, which are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0. These nine characters are called significant figures because they denote some thing. The character 0 is called either zero, cipher, or naught, and stands for nothing, because it is used to fill vacant spaces. When a figure stands alone or in the first place at the right of a number it represents one or more units of the first order. Units of the first order are called simply units and the place they occupy is called units place. The Arabic system is so called because its characters are said to have been introduced into Europe from Arabia, but investigation has proven that this system originated in India. There are two systems of notation in use, Arabic and Roman. The Arabic will be used in all of our arithmetical calculations. Ten is also represented by the figure 1, but the 1 is made to occupy the second place from the right by writing 0 in units place. When a figure stands in the second place from the right it represents one or more units of the second order. Units of the second order are called tens and the place they occupy is called the tens place. Tens and units. The numbers between 10 and 20, 20 and 30, etc., are expressed by representing the tens and units of which they are composed. Read the following numbers: 93, 86, 72, 67, 75, 86, 54, 38, 81, 31, 26, 18, 9, 5. Units of the third order, or hundreds. Ten tens are one hundred; it is represented by the figure 1, in the third order, the orders of tens and units each being filled with a cipher. Units of the third order are called hundreds; and the place they occupy is called the hundreds place. HUNDREDS, TENS, AND UNITS. The numbers between 100 and 200, 200 and 300, etc., are expressed by representing the hundreds, tens, and units of which they are composed. One hundred and one unit are one hundred and one.. written.. 101 ..do.... 102 111 ..do.... 120 125 One hundred, two tens, and five units are one hundred and twenty five..do.... Read the following numbers: 210, 235, 267, 430, 520, 487, 624, 537, 873, 923, 365, 852, 721, 654, 570, 604, 786, 290, 147, 863, 540, 462, 756, 659. UNITS OF HIGHER ORDERS. Ten hundreds are one thousand; it is represented by 1, in the fourth order; thus, 1,000. Ten thousands form a unit of the fifth order; thus, 10,000. One hundred thousands is a unit of the sixth order; thus, 100,000, etc. Invariably ten units of any order make a unit of the next higher order. The names of the first nine orders may be learned from the following: 1. The first nine numbers are represented by the figures 1, 2, 3, 4, 5, 6, 7, 8, 9. 2. All other numbers are represented by the combinations of two or more of the ten figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. 3. The numbers that end with 2, 4, 6, 8, or 0, are called even numbers. 4. The numbers that end with 1, 3, 5, 7, or 9 are called odd numbers. 5. The value of a figure is the number of units it expresses. 6. The value of a figure is always local; that is, it depends upon the place it occupies in a number. 7. The number a figure expresses when it stands in units place is called its simple value. 8. The value of a figure is increased tenfold by removing it one place to the left. 9. The value of a figure is decreased tenfold by removing it one place to the right. GROUPING OF ORDERS IN PERIODS. 1. For convenience in writing and reading numbers the different orders are grouped into periods of three orders each. A number is pointed off into periods of three figures each, using commas. 2. The first three orders, units, tens, and hundreds, constitute the first or unit period. 3. The second group of three orders, thousands, tens of thousands, and hundreds of thousands, constitute the second or thousands period. 4. The third group of three orders constitute the third or million period. 5. The periods from the first to the twelfth, inclusive, may be learned from the following: Table of periods. First Second. Third Fourth. Fifth. Sixth table: The grouping of the orders into periods is shown in the following 7. It is plain that each period is composed of units, tens, and hundreds of that period. To write numbers in the Arabic system: 1. Write: Seven hundred and sixty-nine trillion, five hundred and seventy-six billion, four hundred and thirty-two million, two hundred and eighty-three thousand, one hundred and ninety-eight. Octillion. Nonillion. Decillion. |